Squarespace the sponsor of this video around the 1900s we thought
physics was complete yep we know
everything that is until we started
analyzing the colors of the hot glowing
things I mean we who supposedly know
everything can surely explain what's
going on over here right so we tried we
applied what we knew about guitar
strings and hair molecules to try and
explain it what could go wrong what
could go wrong you might already know
this when we when classical physics
tried to explain the black body
radiation curve it was a disaster we
called it the ultraviolet catastrophe
but I never understood what really
happened why did it happen like that I
mean surely you can understand the
mathematical derivations and by the way
if you want to do that there are some
beautiful videos out there and I have
linked them over there you can go
through them but for me the intuition
was intuition was missing for a long
time that is until you probably know
what I'm going to say I read F lectures
and also went through all these amazing
videos
so in this video you and I are going to
ReDiscover intuitively why classical
physics leads ultimately to this ultra
violet catastrophe and so if you're
ready for this let's begin so fan what's
the whole issue with this hot glowing
things well fman reminds us that it's
not just these metals that glow
everything including you me and donkeys
we all glow due to our temperature we
call this the thermal radiation but
what's important is that this glow
contains all the colors of light all the
wavings of light but they don't all get
the same intensity if you draw the
intensity versus wavelength graph as you
can see over here notice that some
wavelengths like say this particular
infrared gets very little intensity but
certain other wavelength like this
visible light gets a lot of intensity
and some other wavelength again like UV
light again gets very little intensity
and the big question was why does the
graph look like this I mean of course
the graph does change with temperature I
mean at lower temperatures the peak is
in the infrared region that's why most
of us are glowing in infrared and if you
have very high temperatures the peak can
go even in the UV and that's why very
hot Stars can actually give out a lot of
ultraviolet and x-rays and stuff but
pretty much the shape stays the same and
of course we've talked about the black
body radiation graph in so much more
detail in our previous video but
whatever I said in the last couple of
minutes if that made sense then you
don't have to go back and watch the
video I mean let's not let's not break
the flow but again the big question was
why does the graph look like this why
does a longer whing get very little
amount of energy and as the whing gets
shorter and shorter the energy increases
Peaks and then for further short
wavelength the energy decreases again
why this particular shape well classical
physics tried to explain it and tried to
predict this and when they did there was
a
disaster so fine one help me understand
why did the classical physics you know
why did the class why did classical
physics break down what was the disaster
help me understand without any equations
and fan says Mahesh what happens when
you pluck a guitar string and I'm like
what you get music and fan asks who ask
but what what causes that music sound
waves but what causes the sound waves
the waves on the string findan but what
kind of waves I'm like oh my God
standing waves yes standing waves guitar
strings or strings of any musical
instrument for that matter create
standing waves these are different from
traveling waves which well they travel
but you know what I find the word
standing wave very weird because wave is
where you have energy moving from one
point to another without the particles
themselves moving but standing wave that
means is stationary motion it's an
oxymoron kind of like Bittersweet or
acting naturally or Internet privacy but
F reminds us that it's called a standing
wave because you get it when you have
two traveling waves in the opposite
direction and they interfere with each
other again check out this cool visual I
did using chat gbd and some video
editing I'm not trying to sell a course
over here but I'm super super happy
about how I was able to code that so I'm
proud of it so yeah it's pretty cool
right but anyways the same thing happens
when you pluck a guitar string there are
traveling waves which hit the end and
since the ends of the string cannot
vibrate they reflect back and they
interfere causing standing waves but is
this the only way that you can set up a
standing wave no the condition for the
standing wave over here is that the ends
of the string must be fixed there are
more than one ways to do that so another
way to set up a standing wave looks like
this look again the ends are fixed and
if you look carefully you can see there
are two loes over here and therefore the
wavelength over here is half of the
wavelength in the first one can you see
that okay and therefore the frequency of
this one must be twice the frequency of
the previous one and that's why the
second one is oscillating faster but at
this point I'm like wait a second wait a
second I'm a little confused and I
always had this confusion why is it that
the frequency must be double like why
should the frequency be higher if the
wavelength is half Why Can't This
oscillate with the same frequency as the
previous one and I don't want equations
again and fan says well again think in
terms of traveling waves uh if you have
a low frequency then automatically look
the wav becomes longer and if you have a
high frequency look automatically the
wavelength become shorter so clearly
just from the animation this beautiful
animation that I coted myself you can
clearly see that um the wavelength and
the frequency have an inverse
relationship but anyways coming back
over here why stop at half the
wavelength I can now have onethird the
wavelength or Thrice the frequency and
so on and so forth there's no limit to
it I can keep on making the waveling
shorter and shorter and shorter the
frequency becoming higher and higher and
higher there's no limit to how much
standing waves I can set up and that's
the kind of standing waves we get over
here but at this point I'm like okay
that's awesome and great and stuff fine
man but what does this have anything to
do with black body radiation and fman
says Mahesh what do you think will
happen if you were to heat up a Hol
metallic cube with a hole and I'm like
what see technically this graph
represents the thermal radiation of a
perfect absorber a perfect black body
Yumi and donkeys are not perfect now
even though if we're not perfect our
thermal radiation pretty much looks like
this it's similar to this even though
it's not exact but this metallic cube
with the hole simulates pretty much a
perfect black body body and I'm like how
well the metallic cube is certainly not
perfect absorber I mean it can reflect
light but if you to shine light into
that hole then that light is pretty much
gone the chances of that light coming
back out is negligible so in that sense
that hole acts like a perfect absorber
huh and so if you were to heat up that
metallic box and look at the radiation
that comes out from the hole just the
hole not the rest of the cube but just
the hole then that thermal radiation
will have a graph that's pretty much
like this and I'm like that's pretty
clever find one but again
what does this have anything to do with
standing waves or guitar strings well
when you do heat up the metallic box the
walls of the container act like tiny
oscillators producing electromagnetic
waves and now mahes what happens when
these electromagnetic waves travels
forward hit the wall and reflect back
and I'm like just like with the guitar
string we get standing waves and Fon
says yes exactly we get electromagnetic
standing waves and just like with the
guitar strings you can now set standing
waves of shorter and shorter wavelength
or higher and higher frequencies there's
absolutely no limit to how short the
wavelength you can go so you can
practically set up almost all the
frequencies of light and now the
radiation that comes out will contain
all these frequencies of light and I'm
like that is awesome because that now
explains why at any temperature a
thermal radiation will contain all the
frequencies of light Yumi and donkeys
are not just giving out infrared we
giving out all the frequencies of light
remember that it's just that we are
giving most of it in the infrared but
now that brings up the question finean
how do we understand or how do we figure
out how much energy each of these
wavelengths or each of these frequencies
get that's the million doll question
that's what the black body radiation
graph is right and it seems a little
hard to do that because there are
infinite waves infinite wavelengths over
here so how do we calculate how much
energy each one gets and fan says well
mahes it does sound hard or anything
sounds impossible initially but of
course if you can break it down and then
build them up step by step it gets
easier kind of like what we do when
also in the description now back to the video so fan I
understand exactly why at any
temperature thermal radiation contains
all the different wavelength of light
because you can set up all the different
wavelength of light standing waves
inside that Cube that makes sense but
now the big question is how do we
compute how much energy each of these
waves get and for that fan says Mahesh
if you were to pour water randomly into
these these test tubes which of them
will get end up with more water and I'm
like what how am I supposed to know
you're pouring it randomly and if the
any test tube will have more water and F
says exactly but what if we didn't have
one set we had not five but billions and
billions of them and we kept pouring
water randomly in all of them and don't
worry we have lots of water well then
some sets will have a lot of water
others will have less in some sets test
tube one will have more water compared
to the others and in some other test
tubes test tube 3 will have more water
compared to the other it'll be chaotic
and all of that but what if we looked at
things on average meaning if you were to
take let's say all the water in test one
and then added them up and divide by the
total number of test T tubes that we
have that would be the average water in
test tube one and similarly if we did
that for test tube two and test tube 3
and now if we compared the average
amount of water in these three test
tubes mahes which of them would be more
and I'm like I don't know if I have a
feeling that it's all going to be equal
and fan says exactly it would be equal
it's kind of like tossing a coin when
you flip a coin the chances of getting a
heads and tail are the same because
there's no preference now if you flipped
a coin just like 10 to 15 times you
can't tell how many heads and how many
tails you get but if you did it like a
trillion amount of times you can be sure
because there's no preference half of
them would be heads and half of them
would be Tails statistics the same
thing's happening over here because
there's no particular preference for any
one of the test tubes the chances of any
of the test tube being filled to any
level is pretty much the same and
therefore when you do this like billions
and trillions of times and you average
it out on average the amount of water
per test tube will be pretty much the
same and we can now build a theorem out
of this on average water will always be
equally divided among all the different
test tubes and we'll call this the equip
partition theorem and I'm like wait a
second is there a theorem for water in
test tubes and fan says no silly that's
just a analogy it's a powerful analogy
though but now consider your room full
of air molecules these molecules can
move in three directions independently
along the X along the Y and along the
Zed very a second fin one can molecules
also move diagonally well diagonal
motion can be resolved as a combination
of the motion along these three axes but
anyways just like with the test tube
some molecules over here will move fast
some would move slow and some would have
more energy in the horizontal axis
others would have more energy in the
vertical axis but since there's no
preference for any axis on average the
energy has to be equally distributed
among the three directions that's what
the equip partion theorem says on
average energy not water is equally
distributed among various degrees of
freedom degree of freedom is a fancy way
of describing the different ways in
which energies can be stored so for air
molecules the kinetic energy can be
stored in these three different axes and
so there are three degrees of freedom
and so according to equipartition
theorem if the average energy for each
degree of freedom is e not then the
total energy should be three eot makes
sense right and by the way if we had say
diatomic molecules Like Oxygen molecules
for example then there would more
degrees of freedom and at this point I'm
like wait a second find them we're still
in three dimensions how can you have
more than 3° of Freedom well now along
with the m along the three axis they can
also spin that's additional ways of
storing energy in fact they can spin in
a couple of independent ways now and
total therefore they get five degrees of
freedom so in this particular case if on
average the energy per degree of freedom
is e not then the total average energy
would be 5 e makes sense right now so
far this is all statistics and we did
this purely intuitively right but bolman
and Maxwell took this to the next level
they figured out that that average
energy per degree of freedom is
proportional to the absolute temperature
of the system this was huge because for
a long time we didn't really know what
temperature meant but now we do the
temperature is a direct measure of the
average kinetic energy of these
molecules this was such a huge Discovery
back then because this meant that
thermal energy was not a new form of
energy at all it's the same old kinetic
energy but at a much more microscopic
level which was like truly mind bloging
it it took thermodynamics to the whole
new level temperature which was a
macroscopic quantity got connected to
the kinetic energy of these microscopic
particles and what was the connection
bolman's
constant now of course since we're not
diving deep into the mathematics of it
we don't really have to worry too much
about it it's insane that's why I just
wanted to share that it's it's
incredible um but what matters for us
the key take over over here is that the
equip partition theorem says that on
average energy would be equally div
divided among the different degrees of
freedom so if equip partition theorem
works for gas molecules and water in
test tubes it should also work for our
electromagnetic standing waves over here
because at the end of the day it's a
statistical theorem right so over here
look each standing wave is an
independent way of storing thermal
energy and therefore each particular
wavelength represents a degree of
freedom and so Fon says mahes now apply
what you learned about equip partion
theorem over here and tell me what you
get and I'm like okay so according to
equip partion theorem on average each of
these wavelength should get the same
amount of energy let's say e not but
wait a second wait a second wait a
second remember that we can set up any
number of shorter and shorter
wavelengths There's No Limit we can just
set them up so that means there are
infinite degrees of freedom and if each
degree of Freedom gets e not then the
total energy will be
Infinity right it's this this is the
problem isn't it and fan says exactly
this is the Crux of the problem the
problem is there is no limit to how
short you can get your wavelengths over
here there is no limit to that at all
you can just keep on setting shorter and
shorter wavelengths or higher and higher
frequencies and according to equip
partition theorem each of them on
average should have the same amount of
energy so it's kind of like having
infinite test tubes with you it doesn't
matter how much average water there is
in each test tube if there are infinite
test tubes the total amount of water
would be infinity and that's the problem
with classical physics it predicts that
your black bodies at any temperature
because regardless of what the value of
e not is the total energy output should
be Infinity but wait a second wait a
second I understand that this is a
disaster clearly we're getting
Infinities everywhere but shouldn't the
graph be a horizontal line over here
because each uh wavelength each degree
of Freedom gets the same amount of
energy right why is it a curve I don't
see a an ultra violet catastrophe over
here I do see a catastrophe but why is
it an ultravet catastrophe well fan says
this is the last piece of the puzzle and
this is the one that took me a long time
to understand but the key idea the last
piece is the standing string waves are
onedimensional but the standing
electromagnetic waves in that metallic
Cube they are
three-dimensional and I'm like okay how
does that change things well in 1D
there's only one way to set up each
wavelength but turns out in higher
Dimensions there are more than one ways
to set up each wavelength and it's
harder to visualize this in 3D so let's
look at it in 2D again this is something
that I coded myself so but anyways okay
so if you look at the longest wavelength
which is analogous to what we see over
here in 1D then that looks like this and
again there's only one way to set this
up okay however if you were to go for
the next higher frequency you can set it
up this way or you can set it up this
way these are the two ways in which you
can have two loes look at that so since
there are two independent ways to set up
this particular standing waves there are
two degrees of freedom for this
particular wavelength not one and for
even shorter wavelength since you get
more more loes you can even set them up
in multip even more ways and therefore
you get even more degrees of freedom
over
there in short as the frequency
increases unlike in 1D where each of
these vs only have one degree of freedom
because there's only one way to set it
up in higher Dimensions there are are
more and more ways to set them up and as
the frequency increases the number of
ways in which you can set up these
standing waves also increases and so the
number of degrees of freedom
increases at this point I'm like f one
it's I kind of get this but it's still
hard to wrap my head around this and fan
says okay I'm going to take a slightly
more mathematical approach but we'll
keep the math oversimplified so that we
can just talk we can get the intuition
out of it in how many ways can a single
variable take the values of say 1 five
and 8 and I'm like there's only one way
to do that x = 1 x = 5 and xal 7 in each
case there's only one way to set that up
but what if you had a sum of two
variables ooh now there are multiple
ways to set it up for one you can set it
up as 1 Z or 01 both give you a sum of
one but for five which is a higher
number there are more ways to set it up
you can make it one 5 Z or you can make
it four one and so on and so forth and
if you go even higher numbers there are
even more ways to set it up so you can
see by adding two variables the number
of ways in which you can set up
increases and if you had three variables
it would be even more this is kind of
analogous to what's going on again the
math is oversimplified over here the
actual math will involve squares and
square roots and whatnot but at the end
of the day this is what happens when you
add more Dimensions this is why as the
frequency value increases there are more
ways to set up those standing waves in
two and three dimensions and the number
of degrees of freedom just keeps
increasing so if you go back to to our
standing waves in the cube it makes it's
it's more appropriate to think that the
low frequency infrared waves they have
less degrees of freedom say about 10
just to give some number the higher
frequency visible light has more degrees
of freedom say 100 and the ultraviolet
frequency which is even higher has even
more degrees of freedom say a thousand
and according to equip partition theorem
each degree of freedom is supposed to
get e not so who ends up getting most of
the energy still the total energy is in
Infinity like we saw before because
there are infinite degrees of freedom
but within that you can still see the
higher frequency waves win they are the
ones that end up with most of the energy
and so what does the graph look like
it's not a horizontal line but because
there are more degrees of freedom per
frequency at higher frequencies the
graph ends up becoming a curve and this
therefore according to classical physics
not only are you radiating infinite
energy at any temperature but most of
that energy is coming out as high
frequency not really Ultra void but
actually x-ray because x-rays has the
highest frequencies I mean you can have
gamar Ray but gamar Ray are quantum
anyways um but but you get the point the
point is classical physics predicts
ultraviolet catastrophe and by the way
this term ultraviolet catastrophe was
termed was coined much later but anyways
if we zoom out what is now the intuition
behind why classical physics predicted
ultraviolet catastrophe well you have to
put three things together first you can
simulate a perfect black body as a whole
in a metallic Cube where you can set up
up lots and lots of standing waves
there's no limit to how short the
wavelength can be and therefore you have
infinite degrees of freedom second
according to equ partition theorem each
of these degrees of freedom should get
on an average the same amount of energy
which automatically tells you that
you'll have infinite total energy but
finally the third piece is that the
number of degrees of freedom actually
increases with frequency because we're
not in 1D but we are in three dimension
putting it all together gives us that
disastrous result which predicts that
everything should radiate Infinite
Energy mostly in high frequency Ultra or
x-rays so if this all made sense and I
really hope it did the final question
for us is how does quantization solve
all for this what is the logic behind it
well that's coming up in the next video
but till then if you miss my previous
video you can watch here see here bye